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# Solving hairy proportions

Video transcript

We have the proportion x minus
9 over 12 is equal to 2/3. And we want to solve for the x
that satisfies this proportion. Now, there's a bunch of
ways that you could do it. A lot of people, as soon as
they see a proportion like this, they want to cross multiply. They want to say,
hey, 3 times x minus 9 is going to be
equal to 2 times 12. And that's completely
legitimate. You would get-- let
me write that down. So 3 times x minus 9
is equal to 2 times 12. So it would be
equal to 2 times 12. And then you can
distribute the 3. You'd get 3x minus
27 is equal to 24. And then you could
add 27 to both sides, and you would get-- let
me actually do that. So let me add 27 to both sides. And we are left with 3x is
equal to-- let's see, 51. And then x would be equal to 17. And you can verify
that this works. 17 minus 9 is 8. 8/12 is the same thing as 2/3. So this checks out. Another way you could do that,
instead of just straight up doing the cross multiplication
, you could say look, I want to get rid of this 12
in the denominator right over here. Let's multiply both sides by 12. So if you multiply both sides
by 12, on your left-hand side, you are just left
with x minus 9. And on your right-hand
side, 2/3 times 12. Well, 2/3 of 12 is just 8. And you could do the
actual multiplication. 2/3 times 12/1. 12 and 3, so 12
divided by 3 is 4. 3 divided by 3 is 1. So it becomes 2 times
4/1, which is just 8. And then you add
9 to both sides. So the fun of algebra
is that as long as you do something that's
logically consistent, you will get the right answer. There's no one way of doing it. So here you get x is
equal to 17 again. And you could also-- you could
multiply both sides by 12 and both sides by 3, and then
that would be functionally equivalent to cross multiplying. Let's do one more. So here another proportion. And this time the x
in the denominator. But just like before, if we
want we can cross multiply. And just to see where cross
multiplying comes from, it's not some voodoo,
that you still are doing logical
algebra, that you're doing the same thing to
both sides of the equation, you just need to
appreciate that we're just multiplying both sides
by both denominators. So we have this 8 right over
here on the left-hand side. If we want to get rid of
this 8 on the left-hand side in the denominator, we can
multiply the left-hand side by 8. But in order for the
equality to hold true, I can't do something
to just one side. I have to do it to both sides. Similarly, if I want
to get this x plus 1 out of the denominator,
I could multiply by x plus 1 right over here. But I have to do
that on both sides if I want my equality
to hold true. And notice, when you
do what we just did, this is going to be equivalent
to cross multiplying. Because these 8's cancel
out, and this x plus 1 cancels with that x
plus 1 right over there. And you are left with
x plus 1 times 7-- and I can write it as 7 times x
plus 1-- is equal to 5 times 8. Notice, this is
exactly what you would have done if you had
cross multiplied. Cross multiplication
is just a shortcut of multiplying both sides
by both the denominators. We have 7 times x plus
1 is equal to 5 times 8. And now we can go and
solve the algebra. So distributing the 7, we
get 7x plus 7 is equal to 40. And then subtracting
7 from both sides, so let's subtract
7 from both sides, we are left with
7x is equal to 33. Dividing both sides by 7, we are
left with x is equal to 33/7. And if we want to write
that as a mixed number, this is the same
thing-- let's see, this is the same
thing as 4 and 5/7. And we're done.